![Arithmetic Series
• Definition: The sum of an arithmetic sequence.
• Formulae:
• Sn = (n/2) * (a1 + an)
• Sn = (n/2) * [2a1 + (n-1)d]
• Sn: sum of the first n terms
• Example Problem: Find the sum of the first 20 terms of the
sequence: 2, 5, 8, 11...](https://image.slidesharecdn.com/1-240527201911-93de839c/85/sequence-and-series-powerpoint-in-math-assinment-5-320.jpg)
sequence and series powerpoint in math assinment
- 1. Sequences and Series Students Name: Faisal Buradha 224009217 Nawaf Naif Alghanim 224009114 Abdulrahman Almulhim: 224015805 Abdulaziz Alfuwaires 224011618 Abdulhadi almarhabi 224013200 Supervised by Dr. Muhammad Nur
- 2. Introduction What are Sequences? Definition: An ordered list of numbers following a specific pattern or rule. Examples: 2, 4, 6, 8... OR 1, 1, 2, 3, 5, 8... What are Series? Definition: The sum of the terms in a sequence. Examples: 2 + 4 + 6 + 8 + ... OR 1 + 1 + 2 + 3 + 5 + 8 + ... Importance: Sequences and series are fundamental in mathematics and have applications in various fields like finance, physics, and computer science.
- 3. Types of Sequences • Arithmetic Sequences: • Definition: Each term is obtained by adding a constant value (common difference) to the previous term. • Formula: an = a1 + (n-1)d • an: nth term • a1: first term • d: common difference • Example: 3, 7, 11, 15... (d = 4) • Geometric Sequences: • Definition: Each term is obtained by multiplying the previous term by a constant value (common ratio). • Formula: an = a1 * r^(n-1) • an: nth term • a1: first term • r: common ratio • Example: 2, 6, 18, 54... (r = 3) • Other Sequences: Fibonacci sequence, harmonic sequence
- 4. Finding the nth Term •Arithmetic Sequences: Reiterate the formula and provide an example problem: "Find the 10th term of the sequence: 5, 8, 11, 14..."Geometric Sequences: Reiterate the formula and provide an example problem: "Find the 6th term of the sequence: 1, 3, 9, 27..."
- 5. Arithmetic Series • Definition: The sum of an arithmetic sequence. • Formulae: • Sn = (n/2) * (a1 + an) • Sn = (n/2) * [2a1 + (n-1)d] • Sn: sum of the first n terms • Example Problem: Find the sum of the first 20 terms of the sequence: 2, 5, 8, 11...
- 6. Geometric Series • Definition: The sum of a geometric sequence. • Formula: • Sn = a1 * (1 - rn) / (1 - r) • Sn: sum of the first n terms • Example Problem: Find the sum of the first 8 terms of the sequence: 1, 2, 4, 8...
- 7. Infinite Geometric Series • Definition: A geometric series with an infinite number of terms. • Convergence: An infinite geometric series converges (has a finite sum) only if the absolute value of the common ratio (|r|) is less than 1. • Formula: • S = a1 / (1 - r) (if |r| < 1) • S: sum of the infinite series • Example Problem: Find the sum of the infinite series: 1 + 1/2 + 1/4 + 1/8...
- 8. Sigma Notation • Introduction: A shorthand way to represent sums, especially for series. • Explanation: Σ (sigma) symbol represents summation. • Example: Σn=15 n2 = 12 + 22 + 32 + 42 + 52
- 9. Applications of Sequences and Series • Finance: Compound interest, annuities. • Physics: Motion of falling objects, oscillations. • Computer Science: Algorithms, data analysis. • Nature: Fibonacci sequence in plant growth, population growth models.
- 10. Visualizations • Include visually engaging diagrams or graphs: • Graph of an arithmetic and geometric sequence to illustrate their growth. • Diagram showing the convergence of an infinite geometric series. • Visual representation of the Fibonacci sequence in nature (e.g., spiral in a seashell).
- 11. Conclusion Recap the key concepts of sequences and series. Emphasize their importance and real- world applications. Encourage further exploration and learning about this fascinating area of mathematics
- 12. references 1.Khan Academy (https://www.khanacademy.org/) 2.MathWorld (https://mathworld.wolfram.com/)
- 13. Thank you